Development and performance evaluation of an embedded PI controller for pressure regulation in irrigation emitter testingABSTRACT
Tests to characterize the flow rate-pressure curve of irrigation emitters require control of test pressure to ensure the measurement certainty needed in the tests. This study aimed to develop and evaluate a system for automatic control of pressure in tests to determine the flow rate-pressure curve of emitters. To accomplish this, a proportional-integral (PI) controller was implemented in the Arduino, which based on the measurement of the pressure at the emitter inlet, acted on a frequency inverter. The Arduino proportional-integral (PIA) controller was evaluated in comparison with other controllers or control modes, such as a proportional-integral controller of the frequency inverter (PIF), manual control using a potentiometer (MP) and a manual control using human-machine interface of the frequency inverter (MH). In the tests, three test pressures of sprinkler-type emitters (196.1, 294.2, and 392.3 kPa) and two flow rates, corresponding to nozzle combinations (4.0 × 2.8 and 6.4 × 3.2 mm), were considered. In the evaluation, the following performance indices were considered: coefficient of variation (CV), steady-state error (SSE), integral of absolute error (IAE), integral of squared error (ISE), integral of absolute time-weighted error (ITAE), rise time (tr) and settling time (ts). PIA showed the best performance in almost all evaluations and performance indicators, except for CV, for which PIA and PIF controllers had the highest pressure variations, with a maximum CV of 1.7 %. Based on all indicators evaluated indicators, it is concluded that the PIA controller was the most effective alternative for controlling the test pressure of emitters.
Keywords:
PI controller; irrigation engineering; process control; sprinkler
Introduction
The performance of pressurized irrigation systems depends on factors such as climatic conditions, irrigation management, and a design that considers careful selection of equipment, according to technical specifications. The characterization of irrigation equipment through laboratory tests is fundamental (Lima et al., 2003), as it enables the improvement of existing equipment and systems, as well as the development of new products.
Emitter characterization tests are conventionally performed manually, necessitating the supervision of qualified and trained technicians, which requires time and financial capital. Furthermore, instabilities in the test conditions, such as pressure instability, can affect the quality of the results obtained in the tests due to uncertainties in monitoring variables that influence the process (Rocha et al., 2017).
The use of automated control systems, properly designed and validated, provides not only stability of test conditions and a reduction in human error, but also optimizes operating time, thereby enhancing the quality of the results generated by the tests. The efficiency promoted by the proportional-integral-derivative (PID) controller has enabled improvements in tests carried out on test benches, as well as in the collection data (Pinto et al., 2013; Rocha et al., 2017; Saretta et al., 2018).
Although there are several forms of process control, PID is simple for adjusting parameters and can be applied in most control systems (Åström and Hägglund, 2001; Barreiros et al., 2021; Sung, et al., 1995), primarily when the mathematical model of the plant is not known and analytical design methods cannot be used (Basilio and Matos, 2002). The PID controller can provide the most effective correction of instabilities arising from the power grid or other sources that cause uncertainties in the emitter characterization data.
The present study aimed to develop and evaluate an automatic pressure control system applied to tests to determine the flow rate-pressure curve of emitters, comparing the performance of the automatic control mode by means of the developed proportional-integral (PI) controller and proportional-integral controller of the frequency inverter, the manual controllers by means of potentiometer and the manual control using human-machine interface of the frequency inverter (MH).
Materials and Methods
Study site and test bench
The study was conducted in Seropédica, Rio de Janeiro state, Brazil (22°45’21" S, 43°40’28" W, altitude 33 m).
The test bench was composed of the following components (Figure 1): motor-pump set (WEG - 5 hp) — responsible for pressurizing the test bench; frequency inverter (CFW-10) — controls the motor-pump set's rotation, consequently the water pressure; pressure transducer (PSI.420 - ZURICH) — measures the test pressure; multi-jet water meter with pulsed output (CIASEY - 1P/10L) — measures the sprinkler flow rate; sprinkler-type emitter — device to be tested; three-phase energy meter (SM-3W lite) — monitors the parameters of the electrical network; and a microcomputer, used to develop and implement the control logic, adjust the test pressures, and store and process the collected data.
Development of the automatic pressure control system
The developed automatic pressure control system utilizes a PI controller, as shown in Eq. (1), within a closed-loop control system (Freitas et al., 2024).
where u(t) is the controller output, Kp the proportional gain, Ti the integrative time, s; dt the infinitesimal time, plus s; and e(t) the active error, m.
The automatic pressure control system was implemented using the Arduino Mega 2560 prototyping platform. This control hardware features an ATmega 2560 microcontroller, 256 kbyte flash memory, analog and digital inputs and outputs, 8-bit pulse width modulation (PWM) outputs, a serial/USB interface for programming and communication with a computer, and support for the C++ programming language. Arduino was also used to collect the data obtained in the tests.
The block diagram in Figure 2 presents the programming logic developed for the Arduino PI controller. Pressure measurement is performed every second, resulting from the sampling of 1,000 values from the analog-to-digital converter, and converted to pressure using the pressure transducer calibration equation. The output of the controller is converted into a PWM signal, which acts on the frequency inverter, changing the motor-pump set's rotation and consequently the water pressure. Constraints were implemented on the controller so that the PWM performance range was met (0 to 255 bytes). The complete controller circuit is shown in the electrical diagram in Figure 3.
Flowchart of the programming used for proportional-integral controller implementation. Kp = proportional gain; Ti = integrative time; i = repetition of read; P(i) = pressure read; Pm = mean pressure; e = active error; S = controller output; PWM = pulse width modulation.
The integral of the error was determined by numerical integration, considering the rectangle method (Eq. (2)). Since the interval between readings was 1 s, the embedded discrete PI controller was simplified according to Eq. (3).
where n is the current reading, and Δt is the interval between readings, s.
Since the Arduino logic level ranges from 0 to 5 V and the frequency inverter operates from 0 to 10 V, the PWM output was amplified by a factor of 2 and connected to the frequency inverter control interface, as shown in Figure 3.
Performance evaluation of controllers and control modes of the test pressure of emitters
Four different test modalities were evaluated: a) tests with a manual control using a potentiometer (MP); b) tests with a manual control using human-machine interface of the frequency inverter (MH); c) tests with a proportional-integral controller of the frequency inverter (PIF); and d) tests with PIA. For each modality, three test pressures (196.1, 294.2, and 392.3 kPa) and two combinations of nozzles for the emitter (4.0 × 2.8 and 6.4 × 3.2 mm) were evaluated. Each test lasted 10 min, and five repetitions were performed for each pressure tested.
The mean voltage of the phases in the electrical network was measured every 1 min to determine the coefficient of variation (CV). At the end of the tests, Pearson's correlation coefficient (r) between the CV of the electrical network voltage and the CV of the pressure was determined (Table 1) in order to check whether the variations of the electrical network influenced the results.
Coefficient of variation of the electrical network voltage and Pearson's correlation coefficient (r) for the four test modalities and two combinations of nozzles used.
The tests with MP were carried out with a 10 kΩ potentiometer as an interface for adjusting the output frequency of the frequency inverter. Before starting the test, the potentiometer was adjusted until the pressure at the emitter inlet, monitored by the manometer, was equal to the test pressure.
In the tests with manual control by the MH, the test pressure was adjusted via the human-machine interface keys, present on the frequency inverter, whose purpose was to evaluate whether the use of the potentiometer could generate instabilities in the frequency of the inverter and consequently in the test pressure. The pressure adjustment procedure was the same as that adopted in the tests with manual control by potentiometer.
For the tests with the PI controller of the frequency inverter, the PID regulator function, present in the equipment itself, was enabled. The parameterization of the frequency inverter for using the PI controller was carried out according to the instructions in the user manual (Table 2). The setpoint was sent to the frequency inverter using the P525 parameter (digital setpoint). The values used in the P525 parameter were obtained through previous calibration, where increasing values for this parameter were defined and the pressure measured for each nozzle combination (Table 3).
To evaluate the performance of the controllers and control modes, the following parameters were considered: CV and steady-state error (SSE) of the pressure head; integral of absolute error (IAE), integral of squared error (ISE), and integral of absolute time-weighted error (ITAE) (Barreiros et al., 2021; Mohamed et al., 2023), rise time (tr) and settling time (ts), which correspond to the time required for the controller to reach the setpoint and the time from which the controller maintained the pressure with variations of less than 5 %, respectively. The other indicators were calculated according to Eq. (4) to Eq. (7). The integrals were calculated numerically using the rectangle method, with data collection intervals of 1 s.
where Pset is test pressure (setpoint), kPa, γ the density of water, kN m−3; t the time, s; and Pm the mean pressure during the test, disregarding the settling time, kPa.
All indicators were subjected to analysis of variance (ANOVA) at a 5 % significance level. After performing the analysis of variance, variables that showed statistical differences were subjected to a means comparison test (Tukey test) at a 5 % significance level. The analyses were carried out using the statistical analysis and experimental design software SISVAR (Ferreira, 2019).
PIA and PIF controller tuning
Gains by the PI controller were determined using the closed-loop tuning method of Ziegle and Nichols (Basilio and Matos, 2002). This method was chosen because the system's dynamic response was not suitable for the open-loop tuning method. The critical gains (Kcr) and their respective critical periods (Pcr) were determined based on the following procedures: a) the integral and derivative actions were reduced to their minimum effect (Ti = ∞); b) using only the proportional action, a gain was introduced into the system and the response curve generated by it was observed; c) the proportional gain was increased until the system's output curve generated a sustained oscillation of constant amplitude, as shown in Figure 4, and the value of the critical gain was determined; d) from the sustained oscillation generated, the Pcr was determined; and e) with Kcr and Pcr, the values for the parameters Kp and Ti were calculated, as shown in Table 3.
Sustained oscillation in response to proportional action for 294.2 kPa test pressure and 2.8 × 4.0 mm nozzle.
Results
Performance of PIA and PIF controllers based on integrative indicators and transient response
The integrative performance indicators showed that the PIA controller obtained similar or superior performance for the IAE, ISE and ITAE indices for most of the evaluations, for the tests using both the 4.0 × 2.8 mm nozzles and the 6.4 × 3.2 mm nozzles (Table 4), except for the ISE index, relative to the pressure of 392.3 kPa and the 4.0 × 2.8 mm nozzles, for which the PI controller of the frequency inverter resulted in better performance. The controllers’ responses to the pressures tested and to the two nozzle combinations used are presented in Figure 5A and B. During process control, the Arduino PI controller output did not become saturated (Figure 6A and B).
Transient response of the Arduino proportional-integral (PIA) controller and proportional-integral controller of the frequency inverter (PIF): A) 4.0 × 2.8 mm nozzles; B) 6.4 × 3.2 mm nozzles.
Arduino proportional-integral (PIA) controller output: A) 4.0 × 2.8 mm nozzles; B) 6.4 × 3.2 mm nozzles.
For the tests using the 4.0 × 2.8 mm nozzles (Table 5), it was observed that the PIF controller yielded better performance of the tr index for the pressures of 294.2 and 392.3 kPa, while the PIA controller showed better performance for the pressure of 196.1 kPa. This indicates that the response curves of the controllers, for these pressures, reached the desired setpoint values faster, as is shown in Figure 5A. For ts, the PIF also obtained better performance for pressures of 294.2 and 392.3 kPa, requiring less time for the controlled pressure signal to reach a variation range of up to 5 %.
Performance indices based on the transient response of the PIA and PIF controllers, for the nozzles tested.
For the tests using the 6.4 × 3.2 mm nozzles, it was not possible to compare the tr and ts indices between the controllers because the PIF controller did not reach the desired setpoint values for the tested pressures, as also observed in Figure 5B. This fact can be attributed to the mode of transferring the setpoint to the frequency inverter, as the parameter responsible for this task uses input values in percentages. Therefore, it is necessary to conduct additional tests to correlate the test pressures with the corresponding percentage values to be inserted in this parameter.
The tr indicator is correlated with ISE (Table 4), as observed in the tests at a pressure of 196.1 kPa and with 4.0 × 2.8 mm nozzles, where the PIA controller yielded the lowest values for tr and ISE. Similarly, the PIF controller, operating at a pressure of 392.3 kPa, yielded a lower ISE value and a shorter tr, demonstrating the intrinsic relationship between these indicators.
Evaluation of control modes on stability and accuracy of pressure adjustment
Analysis of the CV results (Table 6) found that the PIF and MP control modes yielded similar results in all tests, considering a significance level of 5 %. On the other hand, the MH and PIF control modes did not show a defined behavior in relation to the test pressures and nozzle combinations. The highest CV values were observed in the MH control mode for a test pressure of 294.2 kPa, both for the 4.0 × 2.8 mm nozzles and the 6.4 × 3.2 mm nozzles (1.38 and 1.67 %, respectively).
Results of the coefficients of variation (CV) and steady-state error (SSE) for the nozzles tested.
When analyzing the SSE, it was observed that the PIA controller achieved the best performance for virtually all test pressures and nozzle combinations, with the highest SSE of 0.002 m, providing evidence of the PIA controller's excellent performance. The other control modes did not exhibit stable behavior between the tests, oscillating the performance between the treatments in terms of performance. The control modes with performance statistically equal to the PIA were: MH for 196.1 kPa pressure and 6.4 × 3.2 mm nozzles; MP for the 294.2 kPa pressure in both nozzle combinations and 392.3 kPa pressure for the 6.4 × 3.2 mm nozzles; and MH and PIF for the 392.3 kPa pressure and 4.0 × 2.8 mm nozzles. The control modes that had the highest SSE for the 4.0 × 2.8 mm nozzles were MH, for pressures of 196.1 and 294.2 kPa, and MP, for a pressure of 392.3 kPa. For the 6.4 × 3.2 mm nozzles, the PIF controller exhibited the worst performance at all evaluated pressures, as shown in Figure 5B.
Discussion
For the test pressure of 196.1 kPa and 4.0 × 2.8 mm nozzles, the PIA controller has an ITAE value approximately 4.5 times lower than that of the PIF, which shows better pressure stability after reaching the desired setpoint value. For the same pressure, the values of the IAE indicator were lower for PIA than for PIF, indicating that this controller exhibits a satisfactory transient response and good damping of the response signal. It is worth noting that, in the context of sprinkler characterization tests, pressure stability is the most critical factor, as it can significantly impact the quality of the tests (Saretta et al., 2018).
Although the minimization of ISE shows faster responses for the system (Jandre et al., 2002), since this performance index penalizes larger errors (Barreiros et al., 2021), which usually occur at the beginning of the response, it causes the controlled signal to have greater overshoot, in addition to compromising the stability of the system. The result is oscillations in the response signal and, consequently, higher values for ITAE. Another important point that helps to support this statement is related to the calculation of ITAE (Eq. (5)), since, through time weighting, errors that occur at later times during the test will be more severely penalized than errors that occur at the beginning of the controller's action (Pinto et al., 2013).
From a practical point of view, the PIA and PIF controllers demonstrated satisfactory performance in terms of pressure stabilization time, as this time period was consistently less than 1 min for all tests. According to the normative recommendations of ISO 9261 (ISO, 2004), it is recommended to wait 3 min before carrying out the flow rate measurement.
From a practical point of view, the CV values recorded in all test modalities for the two nozzle combinations used show acceptable variations in test pressures, with values below 4 %, as determined by ISO 15886-3 (ISO, 2021). The reduced CV values obtained may be related to the use of the frequency inverter, which attenuates the oscillations of the electrical network, as found by Saretta et al. (2018) and Rocha et al. (2017), as can be confirmed by the low values of Pearson's correlation coefficient presented in Table 1.
Among the manual pressure control modes, it is observed that, in general, MH showed the most significant instabilities at the test pressures, indicating that this control mode has difficulty in maintaining the test pressures at the desired setpoint values. The study conducted by Rocha et al. (2017) also recorded instabilities in the manual control of this variable during head loss tests and in tests determining the flow rate-pressure curve. The authors observed an improvement in the adjustment of the test pressure relative to the target value, when using a PID controller applied to tests to determine the flow rate-pressure curve of micro-sprinklers.
The results obtained by the PIA controller for the two nozzle combinations used demonstrate that the test pressures adjusted by this control method are highly accurate achieving the desired setpoint with low SSE values, even as low as 0.1 m, which underscores the effectiveness of the PI controller. The PI controller has the advantage of being less sensitive to noise than the PID controller, due to the absence of derivative action (Mudi et al., 2008; Irshad and Ali, 2018). This is the most used control mode in process control for parameters such as flow, level, pressure, and temperature (Foley et al., 2005).
Although the PIA controller was subjected to the most significant variations in electrical energy during the tests (Table 1), it demonstrated superior performance compared to the other test modalities, as evaluated by SSE. Thus, this method of pressure control is the most recommended. Short-term voltage variations result in significant fluctuations in the Root-Mean-Square voltage amplitudes, which significantly impact the operations of loads (Ferreira et al., 2023). It is worth noting that the use of a frequency inverter can help mitigate the impact electrical network variations on the electric motor's operation (Saretta et al., 2018).
The results show that pressure monitoring is essential to emitter characterization tests, and in addition to pressure variation. Additionally, the deviation between the desired pressure values and the mean pressure, which corresponds to the SSE, requires monitoring. It should be noted that the ISO 15886-3 (ISO, 2021) standard does not specify the SSE for emitter tests, establishing only that the test pressure must vary by up to 4 %.
The MH control mode, for the 4.0 × 2.8 mm nozzles, and the PIF control mode, for the 6.4 × 3.2 mm nozzles, showed pressure variations below the limit as determined by ISO 15886-3 (ISO, 2021), and they could not reach the desired setpoints. Thus, the use of the PIA controller is the most recommended for the two nozzle combinations, as it not only complies with a requirement imposed by the standard but also records the lowest SSE values.
All the control modes evaluated met the requirements for pressure variation during the tests. However, the PIF controller exhibited failures in automatically adjusting the test pressure of the emitters for specific tests, resulting in mean pressures lower than the desired pressures. The manual control modes MP and MH demonstrated strong performance in terms of pressure variation but struggled to adjust the test pressure to the desired value. Thus, the PIA controller demonstrated the best performance in terms of automatically adjusting the test pressure of the emitters, with an adequate CV, a short adjustment, and a mean pressure compatible with the target pressure desired.
Data availability statement
The data used and generated in this study are available upon request to the authors.
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Edited by
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Edited by:
Barbara Janet Teruel https://orcid.org/0000-0002-5102-6716
